So long as an opinion is strongly rooted in the feelings, it gains rather than loses in stability by having a preponderating weight of argument against it. ― John Stuart Mill, The Subjection of Women
Many people have strong emotional belief in a certain view, and no amount of logic can uproot their attachment to it. If anything, logic and arguments only end up strengthening their belief, even if evidence clearly shows it to be completely wrong. This was observed by English philosopher and political economist John Stewart Mill 150 years ago. The Quartz article referred to below elaborates on this phenomenon and explains why you can never get anywhere arguing with them.
At the same time, we are often drawn into arguments, or are in a situation where you need to prove yourself to be right. Neuroscience can come in very handy here, and the Mental Floss article provides eleven strategies you can adopt to make your point. Notably, know your opponent and his beliefs, show empathy and a little emotion, stay calm, use physical cues and adopt a story-telling manner.